My Math Forum Calculating Eigen Values error

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 April 18th, 2010, 12:03 PM #1 Senior Member   Joined: Sep 2009 Posts: 251 Thanks: 0 Calculating Eigen Values error Can someone find my error? The correct eigenvalues, if Matlab is to be believed, for the following matrix are -1, 1, 1, 1. I'm getting 1 and 0. $A = \left[ {\begin{array}{cc} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} } \right]$ The only method I know so far for finding eigenvalues is to find the roots of the characteristic polynomial: $det{A-{\lambda}I_4}$ First substitute and subtract; second I added rows 2, 3, and 4 to row 1; third, factored out (1-lambda) $det{ {\begin{array}{cc} -{\lambda} & 0 & 0 & 1 \\ 0 & 1-{\lambda} & 0 & 0 \\ 0 & 0 & 1-{\lambda} & 0 \\ 1 & 0 & 0 & -{\lambda} \\ \end{array} } } =det{ {\begin{array}{cc} 1-{\lambda} & 1-{\lambda} & 1-{\lambda} & 1-{\lambda} \\ 0 & 1-{\lambda} & 0 & 0 \\ 0 & 0 & 1-{\lambda} & 0 \\ 1 & 0 & 0 & -{\lambda} \\ \end{array} } } =(1-{\lambda})det{ {\begin{array}{cc} 1 & 1 & 1 & 1 \\ 0 & 1-{\lambda} & 0 & 0 \\ 0 & 0 & 1-{\lambda} & 0 \\ 1 & 0 & 0 & -{\lambda} \\ \end{array} } }$ Next, expand along the first column; the first determinant is just the diagonal values multiplied: $(1-{\lambda})\left[ (1)(-1)^{1+1}det{\begin{array}{cc} 1-{\lambda} & 0 & 0 \\ 0 & 1-{\lambda} & 0 \\ 0 & 0 & -{\lambda} \\ \end{array} } + 0 + 0 + (1)(-1)^{4+1}det{\begin{array}{cc} 1 & 1 & 1 \\ 1-{\lambda} & 0 & 0 \\ 0 & 1-{\lambda} & 0 \\ \end{array} } \right]=$ $\hspace{50}(1-{\lambda})\left[ (1-{\lambda})^3-det{\begin{array}{cc} 1 & 1 & 1 \\ 1-{\lambda} & 0 & 0 \\ 0 & 1-{\lambda} & 0 \\ \end{array} } \right]$ Swap columns of the second determinant twice (col2<-->col3, then col1<-->col2 (doesn't change the value of the determinant -- well, it changes the sign twice) to make it diagonal and take the determinant by multiplying the values on the diagonal; then distribute: $(1-{\lambda})\left[ (1-{\lambda})^3-det{\begin{array}{cc} 1 & 1 & 1 \\ 1-{\lambda} & 0 & 0 \\ 0 & 1-{\lambda} & 0 \\ \end{array} } \right]=(1-{\lambda})\left[ (1-{\lambda})^3-det{\begin{array}{cc} 1 & 1 & 1 \\ 0 & 1-{\lambda} & 0 \\ 0 & 0 & 1-{\lambda} \\ \end{array} } \right]=(1-{\lambda})\left[ (1-{\lambda})^3-(1-{\lambda})^2 \right]=$ $\hspace{50}\left[ (1-{\lambda})^4-(1-{\lambda})^3 \right]$ The only roots of that last equation are: $\lambda=0, 1$ I just re-rechecked my work as I was typing it in, too. Matlab, the back of my book, and my professor all say that the roots are 1 and -1 (and my professor doesn't like being asked any type of question which we should already know the answer to, like this one). Thanks for your time and help.
 April 18th, 2010, 12:47 PM #2 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 Re: Calculating Eigen Values error The determinant of the first 3x3 matrix should be $-\lambda(1-\lambda)^2,$ not $(1-\lambda)^3.$
 April 18th, 2010, 01:49 PM #3 Senior Member   Joined: Sep 2009 Posts: 251 Thanks: 0 Re: Calculating Eigen Values error I missed that after looking over this problem about five times in the past two days? Thanks!

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